Monotonicity

Q.

Which of the following function is a monotonic function?

1. f(x) = Sin(x)

2. $f\left(x\right)=\frac{1}{x^2}$

3. $f\left(x\right)=x^3$

4. f(x) = {x} ; where {} is the fractional part function.

Q. Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
$h\left(x\right)$ increases as $f\left(x\right)$ decreases for all real values of $x$ if

1. $a\in (0,3)$
2. $a\in (-2,2)$
3. $a\in (3,\infty )$
4. None of these

Q. Let f:$[0,1]\to R$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, $f\left(0\right)=f\left(1\right)=0$ and satisfies $f"\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,x\in [0,1]$
Which of the following is true for $0<x<1$?

1. $0<f\left(x\right)<\infty$
2. $-\frac{1}{2}<f\left(x\right)<\frac{1}{2}$
3. $-\frac{1}{4}<f\left(x\right)<1$
4. $-\infty <f\left(x\right)<0$

Q. Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
$h\left(x\right)$ increases as $f\left(x\right)$ increases for all real values of $x$ if

1. $a\in (0,4)$
2. $a\in (-2,2)$
3. $a\in [3,\infty )$
4. None of these